lemma numbering like 1.2.2 Lemma

Question

definition numbering like the below output, im getting the default numbering only.

1.1 Lemma 1.2. Proposition

how to do this using in latex

coding:

\documentclass[fleqn]{gsm-l}

\usepackage[]{weblink}

\begin{document}

\setcounter{chapter}{0}

\setcounter{section}{0}

\chapter{LINEAR SPACES}

\begin{proposition}

text

\end{proposition}

 \subsection{subsection1}

text

 \begin{lemma}

text
\end{lemma}
\begin{corollary}
 text
\end{corollary}

 \subsection{subsection2}

text

\begin{remark}

 text

\end{remark}

\end{document} 

Output:

Linear Spaces

Proposition 1.0.1. text

1.0.1. Subsection1. text

lemma 1.0.2. text

corollary 1.0.3. text

1.0.2. subsection2. text

remark 1.0.4. text

Expected output:

Linear Spaces

1.0.1. Proposition%its in small caps% text

1.0.2. Subsection1%its in small caps% text

1.0.3. lemma%its in small caps% text

1.0.4. corollary%its in small caps% text

1.0.5. subsection2%its in small caps%. text

1.0.6. Remark%its in small caps% text

without using \newtheorem \newenvironment \theoremstyle \renewcommand

only \newcommand is allowed

weblink

Answer 1

To dfine your theorem-like structures in a consistent way, you can use the amsthm package; the \swapnumbers option gives the numbering and then the title; to get the theorem head in small capitals, you can easily define a new style:

\documentclass{article}
\usepackage{amsthm}
\swapnumbers

\newtheoremstyle{mystyle}
  {\topsep}
  {\topsep}
  {}
  {}
  {\scshape}
  {.}
  {.5em}
  {}
\theoremstyle{mystyle}
\newtheorem{lem}{Lemma}[section]
\newtheorem{pro}[lem]{Proposition}

\begin{document}
\section{Test section one}
\begin{lem}
Your text goes here.
\end{lem}
\begin{pro}
Your text goes here.
\end{pro}
\section{Test section two}
\begin{pro}
Your text goes here.
\end{pro}
\begin{lem}
Your text goes here.
\end{lem}

\end{document}

After the edit to the question, here's an option subject to the imposed constraints: no \newtheoremstyle, no \renewcommand, no \newenvironment (I don't quite understand why these restrictions, specially the one about not using \newenvironment to define environments!; also, weblink uses \newtheorem and \theoremstyle, so why don't you want to use those?):

\documentclass[fleqn]{gsm-l}
\usepackage{weblink}

\let\lemma\relax
\let\corollary\relax
\let\proposition\relax
\let\remark\relax

\newcommand\lemma{%
  \refstepcounter{subsection}%
  \par\noindent\thesubsection.~\textsc{Lemma}.\itshape}
\newcommand\corollary{%
  \refstepcounter{subsection}%
  \par\noindent\thesubsection.~\textsc{Corollary}.\itshape}
\newcommand\proposition{%
  \refstepcounter{subsection}%
  \par\noindent\thesubsection.~\textsc{Proposition}.\itshape}
\newcommand\remark{%
  \refstepcounter{subsection}%
  \par\noindent\thesubsection.~\textsc{Remark}.\itshape}

\begin{document}

\chapter{LINEAR SPACES}

\begin{proposition}
text
\end{proposition}

\subsection{Test subsection one}

text
\begin{lemma}
text
\end{lemma}
\begin{corollary} 
text 
\end{corollary}

\subsection{Test subsection two}
text
\begin{remark}
text
\end{remark}

\end{document}

Answer 2

\documentclass{article}
\newcounter{lem}
\newenvironment{lem}{\refstepcounter{lem}\thesection.\thelem~\textsc{Lemma}.}{\\}
\newenvironment{pro}{\refstepcounter{lem}\thesection.\thelem~\textsc{Proposition}.}{\\}
\begin{document}
\section{Section 1}
    This is the examples\\
    \begin{lem}
        Your text go here.
    \end{lem}
    \begin{pro}
        Your text go here.
    \end{pro}
\section{Section 2}
    \begin{pro}
        Your text go here.
    \end{pro}
    \begin{lem}
        Your text go here.
    \end{lem}
\end{document}

Answer 3

Try this (or maybe I misunderstood your problem):

\documentclass{article}

\newtheorem{lemma}{Lemma}[subsection] % number within subsections
\newtheorem{propo}[lemma]{Proposition}% reuse number of lemma

\begin{document}

\begin{lemma}{foo}\end{lemma}

\begin{propo}{bar}\end{propo}

\subsection{Subsection}

\begin{lemma}{foo}\end{lemma}
\begin{propo}{bar}\end{propo}
\begin{propo}{bar}\end{propo}
\begin{lemma}{foo}\end{lemma}

\end{document}